Unit 11 of 13

Transformations, Approximations and Monte Carlo

  • 6 lessons
  • Videos 68 to 73

This unit finds the distribution of a function of a random variable, extends the law of large numbers, states the central limit theorem and compares exact, approximate and simulated answers.

The CDF method follows an event through a change of variable, and inverse-CDF sampling turns uniform random numbers into draws from a chosen distribution. The central limit theorem says that the standardized sum of independent and identically distributed variables with finite positive variance converges in distribution to the standard normal. This supports normal approximations but gives no accuracy guarantee at a fixed sample size. Poisson and normal approximations to binomial probabilities are checked against exact values, and Monte Carlo estimates are reported with their standard errors.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Changing Variables and Sampling

  1. Transformations and Inverse-CDF Sampling

    How the CDF method finds the distribution of a function of a random variable for monotone and many-to-one maps, why a density can be unbounded, and how inverse-CDF sampling uses the generalized quantile.

    Video 68

Limit Theorems for Averages

  1. The Law of Large Numbers for Continuous Variables

    How the weak law of large numbers covers IID continuous observations with finite variance, how the Chebyshev bound for a sample mean gives a sufficient sample size, and why the bound is conservative and can fail under dependence.

    Video 69

  2. The Central Limit Theorem

    How the central limit theorem describes the standardized sum of IID variables with finite positive variance through convergence in distribution, how to use the normal approximation for an average, and why heavy tails matter.

    Video 70

Exact, Approximate and Simulated Answers

  1. Poisson and Normal Approximations

    When a binomial probability can be approximated by a Poisson law in the rare-event regime or by a normal law with a continuity correction, and how absolute error, relative error and tail accuracy judge the result.

    Video 71

  2. Monte Carlo Probability and Expectation

    How the Monte Carlo method estimates an expectation or a probability by averaging simulated draws, and what its standard error, the square-root cost of precision and an approximate interval mean.

    Video 72

  3. Choosing Exact, Approximate or Simulated Answers

    How to choose between an exact model probability, an approximation and a simulation, and how to report what approximation error, simulation error and uncertain model inputs each mean.

    Video 73